Additive centralized and Stackelberg DEA models for two‐stage system with shared resources

Data envelopment analysis (DEA) is a nonparametric programming method for evaluating the efficiency performance of decision making units (DMUs) with multiple inputs and outputs. The classic DEA model cannot provide accurate efficiency measurement and inefficiency sources of DMUs with complex internal structure. The network DEA approach opens the “black box” of DMU by taking its internal operations into consideration. The complexities of DMU's internal structure involve not only the organization of substages, but also the inputs allocation and the operational relations among the individual stages. This paper proposes a set of additive DEA models to evaluate and decompose the efficiency of a two‐stage system with shared inputs and operating in cooperative and Stackelberg game situations. Under the assumptions of cooperative and noncooperative gaming, the proposed models are able to highlight the effects of strategic elements on the efficiency formation of DMUs by calculating the optimal proportion of the shared inputs allocated to each stage. The case of information technology in the banking industry at the firm level, as discussed by Wang, is revisited using the developed DEA approach.     

A concept of fuzzy input mix-efficiency in fuzzy DEA and its application in banking sector

Data envelopment analysis (DEA) is a linear programming based non-parametric technique for evaluating the relative efficiency of homogeneous decision making units (DMUs) on the basis of multiple inputs and multiple outputs. There exist radial and non-radial models in DEA. Radial models only deal with proportional changes of inputs/outputs and neglect the input/output slacks. On the other hand, non-radial models directly deal with the input/output slacks. The slack-based measure (SBM) model is a non-radial model in which the SBM efficiency can be decomposed into radial, scale and mix-efficiency. The mix-efficiency is a measure to estimate how well the set of inputs are used (or outputs are produced) together. The conventional mix-efficiency measure requires crisp data which may not always be available in real world applications. In real world problems, data may be imprecise or fuzzy. In this paper, we propose (i) a concept of fuzzy input mix-efficiency and evaluate the fuzzy input mix-efficiency using α – cut approach, (ii) a fuzzy correlation coefficient method using expected value approach which calculates the expected intervals and expected values of fuzzy correlation coefficients between fuzzy inputs and fuzzy outputs, and (iii) a new method for ranking the DMUs on the basis of fuzzy input mix-efficiency. The proposed approaches are then applied to the State Bank of Patiala in the Punjab state of India with districts as the DMUs.     

Two-stage DEA model with additional input in the second stage and part of intermediate products as final output

Data envelopment analysis (DEA) has been proved to be an excellent approach for measuring performance of decision making units (DMUs) that use multiple inputs to generate multiple outputs. In many real world scenarios, inputs or outputs may be shared among various activities. This paper proposes a two-stage DEA model with additional input in the second stage and part of intermediate products as final output. We first discuss the non-cooperative condition in order to determine the upper and lower bounds of the efficiencies of sub-DMUs in different stages. A parametric transformation is described to solve the non-linear programming of the overall cooperative efficiency model. An application is provided.     

Slacks-based measures of efficiency in imprecise data envelopment analysis: An approach based on data envelopment analysis with double frontiers

Data envelopment analysis (DEA) is a mathematical approach for evaluating the efficiency of decision-making units (DMUs) that convert multiple inputs into multiple outputs. Traditional DEA models assume that all input and output data are known exactly. In many situations, however, some inputs and/or outputs take imprecise data. In this paper, we present optimistic and pessimistic perspectives for obtaining an efficiency evaluation for the DMU under consideration with imprecise data. Additionally, slacks-based measures of efficiency are used for direct assessment of efficiency in the presence of imprecise data with slack values. Finally, the geometric average of the two efficiency values is used to determine the DMU with the best performance. A ranking approach based on degree of preference is used for ranking the efficiency intervals of the DMUs. Two numerical examples are used to show the application of the proposed DEA approach.     

Target setting with minimum improving costs in data envelopment analysis: A mixed integer linear programming approach

This study addresses a problem called cost‐minimizing target setting in data envelopment analysis (DEA) methodology. The problem is how to make an inefficient decision‐making unit efficient by allocating to it as few organizational resources as possible, assuming that the marginal costs of reducing inputs or increasing outputs are known and available, which is different from previous furthest and closest DEA targets setting methods. In this study, an existed cost minimizing target setting heuristics approach based on input‐oriented model is examined to show that there exist some limitations. This study develops a simple mixed integer linear programming to determine the desired targets on the strongly efficient frontier based on non‐oriented DEA model considering the situation in the presence of known marginal costs of reducing inputs and increasing outputs simultaneously. Some experiments with the simulated datasets are conducted, and results show that the proposed model can obtain more accurate projections with lower costs compared with those from furthest and closest target setting approaches. Besides, the proposed model can be realistic and efficient in solving cost‐minimizing target setting problem.     

Multiobjective target setting in data envelopment analysis using AHP

In this paper two new target setting DEA approaches are proposed. The first one is an interactive multiobjective method that at each step of the process asks the decision maker (DM) which inputs and outputs he wishes to improve, which ones are allowed to worsen and which ones should stay at their current level. The local relative priorities of these inputs and outputs changes are computed using the analytic hierarchy process (AHP). After obtaining the candidate target, the DM can update his preferences for improving, worsening or maintaining current inputs and outputs levels and obtain a new candidate target. Thus continuing, until a satisfactory operating point is computed. The second method proposed uses a lexicographic multiobjective approach in which the DM specifies a priori a set of priority levels and, using AHP, the relative importance given to the improvements of the inputs and outputs at each priority level. This second approach requires solving a series of models in order, one model for each priority level. The models do not allow for worsening of neither inputs nor outputs. After the lowest priority model has been solved the corresponding target operating point is obtained. The application of the proposed approach to a port logistics problem is presented.     

The voting analytic hierarchy process method for discriminating among efficient decision making units in data envelopment analysis

Making optimal use of available resources has always been of interest to humankind, and different approaches have been used in an attempt to make maximum use of existing resources. Limitations of capital, manpower, energy, etc., have led managers to seek ways for optimally using such resources. In fact, being informed of the performance of the units under the supervision of a manager is the most important task with regard to making sensible decisions for managing them. Data envelopment analysis (DEA) suggests an appropriate method for evaluating the efficiency of homogeneous units with multiple inputs and multiple outputs. DEA models classify decision making units (DMUs) into efficient and inefficient ones. However, in most cases, managers and researchers are interested in ranking the units and selecting the best DMU. Various scientific models have been proposed by researchers for ranking DMUs. Each of these models has some weakness(es), which makes it difficult to select the appropriate ranking model. This paper presents a method for ranking efficient DMUs by the voting analytic hierarchy process (VAHP). The paper reviews some ranking models in DEA and discusses their strengths and weaknesses. Then, we provide the method for ranking efficient DMUs by VAHP. Finally we give an example to illustrate our approach and then the new method is employed to rank efficient units in a real world problem.     

A modified slacks‐based ranking method handling negative data in data envelopment analysis

Performance ranking for a set of comparable decision‐making units (DMUs) with multiple inputs and outputs is an important and often‐discussed topic in data envelopment analysis (DEA). Conventional DEA models distinguish efficient units from inefficient ones but cannot further discriminate the efficient units, which all have a 100% efficiency score. Another weakness of these models is that they cannot handle negative inputs and/or outputs. In this paper, a new modified slacks‐based measure is proposed that works in the presence of negative data and provides quantitative data that helps decision makers obtain a full ranking of DMUs in situations where other methods fail. In addition, the new method has the properties of unit invariance and translation invariance, and it can give targets for inefficient DMUs to guide them to achieve full efficiency. Two numerical examples are analysed to demonstrate the usefulness of the new method.     

基于基尼系数-交叉效率的多属性决策方法

在数据包络分析(DEA)方法的基础上,提出一种基于基尼系数-交叉效率的多属性决策方法,用于解决具有多个投入、产出指标的多属性决策问题.首先,借鉴基尼系数的优化准则构建基尼系数-交叉评价策略模型,从而得到相对唯一的DEA交叉效率矩阵;然后,应用基尼准则计算各个效率值所包含的信息纯度,并借之实现交叉效率矩阵的集结;最后,根据集结结果对决策单元进行排序和择优.所提决策方法不仅能够克服传统DEA交叉效率方法的交叉评价策略选择难的问题,而且能够保证决策过程的客观性和公平性.同时,所提方法还对交叉评价所得的决策信息进行提纯,为科学合理地进行决策提供更多的有效信息.通过对中国各地区的医疗资源配置效率进行实证,验证了所提出方法的有效性和实用性.     

Data envelopment analysis of integer-valued inputs and outputs

This paper addresses DEA scenarios whose inputs and outputs are naturally restricted to take integer values. Conventional DEA models would project the DMU onto targets that generally do not respect such type of integrality constraints. Although integer-valued inputs and outputs can be considered as a special case of ordinal inputs and outputs, the use of that type of models has many drawbacks. In this paper a MILP DEA model that guarantees the required integrality of the computed targets is proposed.     

Improving the discrimination of data envelopment analysis models in multiple time periods

The trade‐offs approach is an advanced tool for the improvement of the discrimination of data envelopment analysis (DEA) models; this can improve the traditional meaning of efficiency as a radial improvement factor for inputs or outputs. Therefore, the Malmquist index – the prominent index for measuring the productivity change of decision making units (DMUs) in multiple time periods that use DEA models with variable returns to scale and constant returns to scale technologies – can be improved by using the trade‐offs technology. Hence, an expanded Malmquist index can be defined as an improved method of a traditional Malmquist index that uses the production possibility set, which could present more discrimination of DMUs, in the presence of the trade‐offs technology. In addition, similar to a traditional Malmquist index, it breaks down into different components. An illustrative example is presented to show the ability of the suggested method of presenting the Malmquist index from a computational point of view.     

Recognizing strong and weak congestion slack based in data envelopment analysis

The common concept of congestion is that a decrease (increase) in one or more inputs of a decision making unit (DMU) causes an increase (decrease) in one or more outputs (Cooper, Gu, & Li, 2001a). So far several congestion approaches have been proposed in DEA (data envelopment analysis) literature by many authors, such as Färe’s et al. (FGL), Brockett’s et al. (BCSW), and Tone and Sahoo’s congestion approaches (Färe et al., 1985, Färe et al., 1994, Brockett et al., 1998, Tone and Sahoo, 2004). Tone and Sahoo’s approach (Tone & Sahoo, 2004) is one of the most robust congestion approaches in DEA literature. Moreover, Tone and Sahoo’s approach has some advantages with respect to FGL and BSCW congestion approaches. However, the proposed approaches have many difficulties to treat congestion. For instance, in the presence of alternative optimal solutions, the approach proposed by Tone and Sahoo is unable to detect congestion (strong and weak). Moreover, in Tone and Sahoo’s approach, all inputs and outputs of decision making units (DMUs) have been considered positive, while in real world, data is often non-negative.In this research, a slack-based DEA approach is proposed to recognize congestion (strong and weak) for the target DMUs. One of the advantages of our proposed approach is capable of detecting congestion (strong and weak) for evaluating the DMUs in the presence of alternative optimal solutions. Other advantage of our research is capable of identifying congesting (strong and weak) DMUs with non-negative inputs and outputs. However in these situations, Tone and Sahoo’s congestion approach is incapable of identifying congestion. Lastly, we apply the approach to the data sets for making comparisons between the proposed approach and Tone and Sahoo’s approach then some conclusions are drawn and directions for future research are suggested.     

Combining DEA with location analysis for the effective consolidation of services in the health sector

One of the most important activities of strategic planning in a health-care system is the effective allocation of scarce resources. Most of such strategies are attempting to create more efficient systems based on better organizational and management structures. Therefore, it is necessary to develop systematic models and evaluation methods that will support a strategic planning process that addresses issues such as the location of services and the effective use of resources such as equipment, funds or workforce. Such modeling approaches need to quantify the effect of changes in the location of providers, the opening or closure of providers, and the dynamic transformations of the services offered at each provider. In this paper we propose a methodology that takes into account health service provider efficiencies based on multiple measures. These efficiencies are then employed to determine health providers' locations and service allocations, which include new services distribution as well as existing services redistribution. This approach employs data envelopment analysis (DEA) and integer programming (IP) location allocation models and can be used as both an immediate evaluation tool and a long-term planning aid.     

A study of the relative efficiency of Chinese ports: a financial ratio‐based data envelopment analysis approach

Abstract: This research examines the relative efficiency of 11 major Chinese ports by using an innovative adopted version of Data Envelopment Analysis (DEA). DEA is a non‐parametric approach to weigh the inputs/outputs and measure the relative efficiency of decision‐making units. This paper adopts an output‐oriented version of DEA based on financial ratios in which no inputs are utilized. The new adopted DEA model provides a rounded judgement on port efficiency taking into consideration multiple financial ratios simultaneously and combining them into a single measure of efficiency. The mathematical model is solved for every port, and the relative efficiency of each port is determined. The results of DEA show that the higher a port's efficiency ratio in relation to the corresponding ratio of another port, the higher the efficiency of this port. Finally, suggestions based on the data analysis are provided for managerial decision makers to improve the areas needed for port operating efficiency.     

Data Envelopment Analysis Based on Choquet Integral

Data envelopment analysis (DEA) is a widely used technique in decision making. The existing DEA models always assume that the inputs (or outputs) of decision‐making units (DMUs) are independent with each other. However, there exist positive or negative interactions between inputs (or outputs) of DMUs. To reflect such interactions, Choquet integral is applied to DEA. Self‐efficiency models based on Choquet integral are first established, which can obtain more efficiency values than the existing ones. Then, the idea is extended to the cross‐efficiency models, including the game cross‐efficiency models. The optimal analysis of DEA is further investigated based on regret theory. To estimate the ranking intervals of DMUs, several models are also established. It is founded that the models considering the interactions between inputs (or outputs) can obtain wider ranking intervals.     

New analytical hierarchical process/data envelopment analysis methodology for ranking decision‐making units

The analytical hierarchical process/data envelopment analysis (AHP/DEA) methodology for ranking decision‐making units (DMUs) has some problems: it illogically compares two DMUs in a DEA model; it is not compatible with DEA ranking in the case of multiple inputs/multiple outputs; and it leads to weak discrimination in cases where the number of inputs and outputs is large. In this paper, we propose a new two‐stage AHP/DEA methodology for ranking DMUs that removes these problems. In the first stage, we create a pairwise comparison matrix different from AHP/DEA methodology; the second stage is the same as AHP/DEA methodology. Numerical examples are presented in the paper to illustrate the advantages of the new AHP/DEA methodology.     

Measurement of overall performances of decision-making units using ideal and anti-ideal decision-making units

Data envelopment analysis (DEA) is a method for evaluating relative efficiencies of decision-making units (DMUs) which perform similar functions in a production system, consuming multiple inputs to produce multiple outputs. The conventional form of DEA evaluates performances of DMUs only from the optimistic point of view. In other words, it chooses the most favorable weights for each DMU. There is another approach that measures efficiency of a DMU from the pessimistic point of view. This approach chooses the most unfavorable weights for evaluation of each DMU. In this paper, we propose to integrate both efficiencies in the form of an interval in order to measure the overall performance of a DMU. The proposed DEA models for evaluation of efficiencies are called bounded DEA models. The proposed approach will be compared using a numerical example. Another example regarding performance evaluation of 50 bank branches in Iranian cities will be presented to demonstrate the advantages, simplicity, and utility of this approach in real-life situations.     

Allocation of shared costs among decision making units: a DEA approach

An issue of considerable importance, both from a practical organizational standpoint and from a costs research perspectives, involves the allocation of fixed resources or costs across a set of competing entities in an equitable manner. Cook and Kress (Eur. J. Oper. Res. 119 (1999) 652) propose a data envelopment analysis (DEA) approach to obtain a theoretical framework for such cost allocation problems. Their approach cannot be used directly to determine a cost allocation among the decision making units (DMUs), but rather to examine existing costing rules for equity. The current paper extends the Cook and Kress (Eur. J. Oper. Res. 119 (1999) 652) approach, and provides a practical approach to the cost allocation problem. It is shown that an equitable cost allocation can be achieved using DEA principles.     

Multi-Output Regularized Feature Projection

Dimensionality reduction by feature projection is widely used in pattern recognition, information retrieval, and statistics. When there are some outputs available (e.g., regression values or classification results), it is often beneficial to consider supervised projection, which is based not only on the inputs, but also on the target values. While this applies to a single-output setting, we are more interested in applications with multiple outputs, where several tasks need to be learned simultaneously. In this paper, we introduce a novel projection approach called multi-output regularized feature projection (MORP), which preserves the information of input features and, meanwhile, captures the correlations between inputs/outputs and (if applicable) between multiple outputs. This is done by introducing a latent variable model on the joint input-output space and minimizing the reconstruction errors for both inputs and outputs. It turns out that the mappings can be found by solving a generalized eigenvalue problem and are ready to extend to nonlinear mappings. Prediction accuracy can be greatly improved by using the new features since the structure of outputs is explored. We validate our approach in two applications. In the first setting, we predict users' preferences for a set of paintings. The second is concerned with image and text categorization where each image (or document) may belong to multiple categories. The proposed algorithm produces very encouraging results in both settings     

A generalized multiple layer data envelopment analysis model for hierarchical structure assessment: A case study in road safety performance evaluation

Data envelopment analysis (DEA) is a powerful analytical research tool for measuring the relative efficiency of a homogeneous set of decision making units (DMUs) by obtaining empirical estimates of relations between multiple inputs and multiple outputs related to the DMUs. To further embody multilayer hierarchical structures of these inputs and outputs in the DEA framework, which are prevalent in today’s performance evaluation activities, we propose a generalized multiple layer DEA (MLDEA) model. Starting from the input-oriented CCR model, we elaborate the mathematical deduction process of the MLDEA model, formulate the weights in each layer of the hierarchy, and indicate different types of possible weight restrictions. Meanwhile, its linear transformation is realized and further extended to the BCC form. To demonstrate the proposed MLDEA model, a case study in evaluating the road safety performance of a set of 19 European countries is carried out. By using 13 hierarchical safety performance indicators in terms of road user behavior (e.g., inappropriate or excessive speed) as the model’s input and 4 layered road safety final outcomes (e.g., road fatalities) as the output, we compute the most optimal road safety efficiency score for the set of European countries, and further analyze the weights assigned to each layer of the hierarchy. A comparison of the results with the ones from the one layer DEA model clearly indicates the usefulness and effectiveness of this improvement in dealing with a great number of performance evaluation activities with hierarchical structures.